On a weighted linear matroid intersection algorithm by deg-det computation

TL;DR

This paper presents a novel algebraic algorithm for the weighted linear matroid intersection problem, leveraging noncommutative determinant degrees, which offers a different approach that can handle sparse matrix representations.

Contribution

It introduces an algebraic algorithm based on noncommutative determinant degrees that aligns with Frank's weight splitting method, enabling efficient handling of sparse matrices.

Findings

Algorithm runs in O(mn^3 log n) time.

Works with sparse matrix representations, skipping Gaussian eliminations.

Provides a linear algebraic perspective on Frank's algorithm.

Abstract

In this paper, we address the weighted linear matroid intersection problem from the computation of the degree of the determinants of a symbolic matrix. We show that a generic algorithm computing the degree of noncommutative determinants, proposed by the second author, becomes an $O(mn^3 \log n)$ time algorithm for the weighted linear matroid intersection problem, where two matroids are given by column vectors $n \times m$ matrices $A,B$. We reveal that our algorithm is viewed as a "nonstandard" implementation of Frank's weight splitting algorithm for linear matroids. This gives a linear algebraic reasoning to Frank's algorithm. Although our algorithm is slower than existing algorithms in the worst case estimate, it has a notable feature: Contrary to existing algorithms, our algorithm works on different matroids represented by another "sparse" matrices $A^0,B^0$, which skips unnecessary Gaussian eliminations for constructing residual graphs.